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International Macroeconomics 5: The Monetary Approach, and Closed Economy RBC Models!

Brendan Epstein, Ph.D. Johns Hopkins University

Contents

1 Preliminaries 1

2 Closed Economy Real Business Cycle Models 4 2.1 Decentralized Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 The Household . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2 The Firm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.3 Closing the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.4 The Bottom Line: Key Equations . . . . . . . . . . . . . . . . . . . . 15 2.1.5 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Centralized/Planning Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Analytical Steady States 18

1 Preliminaries

" Much of the focus of Macroeconomics and International Macroeconomics lies in the analysis of the business cycle component of real variables. The idea behind this focus is gauging stylized facts that characterize aggregate behavior. The second part of this analysis involves developing models to try to understand the driving forces behind these stylized facts.

" The following table presents information to help assess key stylized facts involving international real business cycles. I created this table using Matlab and the previous Lessonís InternationalData.xlsx Öle. I will guide you through the interpretation of this table in the Panopto video called ìOECD Stylized Facts.î Some things to note:

!These lecture notes closely and sometimes literally follow sections from: Epstein, Brendan. Masters Level International Macroeconomics, c# 2013.

ñ The data are at yearly frequency, so to extract the cyclical components I used an HP Ölter with smoothing parameter 6.25.

ñ All the cyclical components are based o§ the natural logarithm of the data, except in the case of net exports. This is because net exports can be negative, so the natural logarithm would be undeÖned.

ñ The statistics presented in each panel are fairly standard for this type of analysis.

" I will give you an overview of how to generate this table using Matlab in the Panopto video titled ìOECD Stylized Construction.î

" Another expositional means of interest for characterizing the behavior of key macro aggregates is impulse response functions. The graphs below show impulse response functions given a 1-unit shock obtained using an AR(p) speciÖcation for key macro aggregates and for each country. I created these graphs using the aforementioned data and Stata. I will guide you through the interpretation of these graphs in the Panopto video called ìOECD IRFs.î

-1 -.

5 0

.5 1

P er

ce nt

D ev

ia tio

n fr

om S

te ad

y S

ta te

0 10 20 30 40 Periods After Shock

US_yIRF Aus_yIRF Aut_yIRF Can_yIRF Fra_yIRF Ger_yIRF Ita_yIRF Jap_yIRF Swi_yIRF UK_yIRF

Output IRFs

-1 -.

5 0

.5 1

P er

ce nt

D ev

ia tio

n fr

om S

te ad

y S

ta te

0 10 20 30 40 Periods After Shock

US_cIRF Aus_cIRF Aut_cIRF Can_cIRF Fra_cIRF Ger_cIRF Ita_cIRF Jap_cIRF Swi_cIRF UK_cIRF

Consumption IRFs

-1 -.

5 0

.5 1

P er

ce nt

D ev

ia tio

n fr

om S

te ad

y S

ta te

0 10 20 30 40 Periods After Shock

US_iIRF Aus_iIRF Aut_iIRF Can_iIRF Fra_iIRF Ger_iIRF Ita_iIRF Jap_iIRF Swi_iIRF UK_iIRF

Investment IRFs

-. 5

0 .5

1 P

er ce

nt D

ev ia

tio n

fr om

S te

ad y

S ta

0 10 20 30 40 Periods After Shock

US_gIRF Aus_gIRF Aut_gIRF Can_gIRF Fra_gIRF Ger_gIRF Ita_gIRF Jap_gIRF Swi_gIRF UK_gIRF

Government Spending IRFs

-1 -.

5 0

.5 1

P er

ce nt

D ev

ia tio

n fr

om S

te ad

y S

ta te

0 10 20 30 40 Periods After Shock

US_nxIRF Aus_nxIRF Aut_nxIRF Can_nxIRF Fra_nxIRF Ger_nxIRF Ita_nxIRF Jap_nxIRF Swi_nxIRF UK_nxIRF

Net Exports IRFs

-1 -.

5 0

.5 1

P er

ce nt

D ev

ia tio

n fr

om S

te ad

y S

ta te

0 10 20 30 40 Periods After Shock

US_hIRF Aus_hIRF Aut_hIRF Can_hIRF Fra_hIRF Ger_hIRF Ita_hIRF Jap_hIRF Swi_hIRF UK_hIRF

Work Hours IRFs

" I will give you an overview of how to generate these graphs using Stata in the Panopto video titled ìOECD IRFs Construction.î

" Having gone over key stylized facts of international real business cycles, the next step is to develop proÖciency in modeling strategies developed to understand the driving forces of these stylized facts. These models are, unsurprisingly, called real business cycle (RBC) models.

" Given the complexity of these models, we will momentarily focus on a closed economy in order to build foundations for intertemporal optimization. Classic references for

RBC models are, for instance, Kydland and Prescott (Econometrica, 1982), and King, Plosser, and Rebelo (Journal of Monetary Economics, 1988). This modeling framework is the one that international (or open economy) real business cycle models (IRBC or IBC) are built on (see, for instance, Backus, Kehoe, and Kydland, Journal of Political Economy, 1992). All told, in Lesson 5 will be going over a series of benchmark RBC setups, and then in Lesson 7 we will develop the benchmark IBC model. This bench- mark model is, in fact, the workhorse model of modern International Macroeconomics upon which analysis of the real economy is based on.

" Boththe RBCandIBCmodels we will be lookingat fall within the categoryof dynamic stochastic general equilibrium models (DSGE). These models are characterized by be- ing cast within a general equilibrium framework and the presence of uncertainty about future states of the economy, where the ìgeneral equilibriumî means that the optimiza- tion of all economic agents is taken into account and all markets clear simultaneously. In addition, these models are cast from a representative agent framework and often assume that economies are inÖnitely lived. The main actors in these models are house- holds (alternatively, ìthe householdî given the ìrepresentative agentî assumption) and Örms (alternatively, ìthe Örm,î again, given the ìrepresentative agentî assumption). Importantly, as far as the household goes, the modeling framework assumes that the entirety of the economyís population is identical, so all individuals are homogeneous, and that there is perfect risk sharing between the population of a country. Moreover, the population is assumed to have a unit mass (which is akin to normalizing the popu- lation to one). All told, given these assumptions the setup of the model economy is the same whether it is interpreted from the vantage point of a representative household or whether it is interpreted from the vantage point of a single representative individual.

2 Closed Economy Real Business Cycle Models

" The main way of setting up a real business cycle model from a decentralized perspective is assuming that households own Örms, but management and ownership are distinct, so households behave as though Örm proÖts at any arbitrary time t, denoted by !t, are given (! is the Greek letter ìcapital piî). In addition, it is generally assumed, albeit without loss of generality, that the household owns the economyís capital stock. This is the Örst framework we will examine. That said, an alternative formulation that yields the same solution involves a structure in which Örms own the economyís capital stock. Whether the household or the Örm owns the capital stock results in the same set of equilibrium conditions, so we will not be going over this second decentralized case, especially since the case in which the household owns the capital stock is the more common speciÖcation.

" Finally, we will see that a third ìequivalentî formulation involves the solution to a centralized/planning problem in which a ìbenevolent social plannerî makes all the decisions for theeconomy. This last formulation is themostwidelyused inInternational Macroeconomic contexts appropriately extended to make the model economy open rather than closed). The reasons for the centralized/planning problem being so widely

used in International Macroeconomics is because it is exceedingly simpler to implement than the decentralized options. In contrast, in a closed economy environment the decentralized and centralized/planning options are pretty much equivalent as far as work e§ort goes.

ñ Of note, politics, political economy, and whatnot are not involved in the central- ized/planning version of things, nor does the equivalence between this framework and the decentralized frameworks imply policy recommendations of any sort as per- taining to socioeconomic systems. The equivalence between all problems is purely of mathematical interest and application.

" In all cases, we assume that the economy is inhabited by a continuum of inÖnitely lived individuals whose mass is normalized to 1 and who are grouped into an aggregate risk sharing household. In turn, there is a representative Örm that produces the Önal good. This good is unique, it can be used towards consumption or investment, and its price is normalized to 1. (Thatís why the model is ìreal:î all prices are in relative terms (relative to consumption), so theyíre real pricesónote, for example, that the wage is also ìa price:î itís the relative price of leisure). Moreover, all non- price variables are normalized by the economyís population. In all frameworks studied all markets are assumed to be perfectly competitive and, for simplicity and for now, there is no government sector.

ñ Let me address in a bit more detail the idea of normalized prices. In the models weíll be developing, there are 5 prices: the price of the Önal good; the price of consumption; the price of investment; the price of labor (the wage); and the price of capital (the rental rate). In any arbitrary time period t let these prices be denoted, in nominal terms, respectively by pY;t, pC;t, pI;t, pw;t, and pR;t. The fact that the ìÖnal goodî can be devoted towards consumption or investment means that the price of this good, as well as the prices of consumption and investment must be the sameóitís just the same good. So,

pY;t = pC;t = pI;t.

Now, letís normalize all prices by the price of the Önal good and give them some new notation where appropriate:

pY;t pC;t

= 1; pC;t pC;t

= 1; pI;t pC;t

= 1; pw;t pC;t

$ wt; and pR;t pC;t

$ Rt.

Note that the price of capital is also referred to as the ìrental rateî of capital. Above, without loss of generality I divided everything by the nominal price of consumption. Whatís on the right-hand side of the equalities/identities is the real price of each variable. So, for instance wt tells you how many units of the consumption good can purchased for each ìunitî of the wage.

ñ Similarly, letís think a bit more about what it means to normalize all non-price variables by the population. Let Pt denote the aggregate population in any arbi- trary period t. And, letís consider two of the aggregate variables weíll be dealing

with: consumption and investment. These are non-price variables. In the no- tation these variables are to be interpreted in per population terms given the normalization adopted for non-price variables. In essence, in the model:

Pt Pt

= 1;

consumption Pt

$ ct (or Ct; notations in the literature vary);

investment Pt

$ It (or it; notations in the literature vary);

and so on and so forth for all non-price variables.

" Finally, note that the representative agent framework we are embarking in implies that all variables should be interpreted as aggregates (e.g., aggregate consumption, aggregate investment, etc.).

2.1 Decentralized Setup

2.1.1 The Household

" The household discounts the future by the subjective parametric discount factor * 2 (0;1), where * is the Greek letter ìbeta.î (The need for a discount factor arises from the intertemporal nature of the framework we will be developing; in essence, as you will see within context * is simply capturing the fact that the future is ìvalued lessî than the present). The household supplies labor nt measured in hours and consumes ct. In addition, the household obtains utility from consumption and leisure, and the householdís total time endowment is normalized to 1 (the framework could also be cast from the vantage point of the household deriving utility from consumption and disutility from work hours); as such, total leisure is given by 1 & nt. In exchange for its labor services, the household obtains the wage rate wt, which is paid by the Örm and taken as given. In addition, the household rents the capital stock to the Örm in exchange for which it obtains the rental rate Rt, which is also taken as given. Finally, the household owns the Örm, but the Örm operates independently from the household so that the household receives proÖts !t from the Örm that are taken as given.

" The household budget constraints are:

ct + It ' wtnt +Rtkt +!t;

that is: consumption (ct) plus investment (It), which amount to total household expendituresóinvestment is used to Önance capital accumulationómust be less than or equal to total labor income (wtnt), total rental income (Rtkt, where kt denotes the economyís capital stock) and Örm proÖts (!t), which every period the Örm gives to the household; and the equation of motion for capital:

kt+1 &kt = It & .kt,

where ., the Greek letterìdelta,î is the capital depreciation rate. This equation implies that there is a one period ìtime to buildî involved in the ìproductionî of capital. In essence (recall that we are referring to physical capital here) this equation says that the capital available tomorrow is equal to the total amount invested today plus the total amount of capital available today net of depreciation (indeed, note that the equation can be rearranged to yield:

kt+1 = It +(1& .)kt.

Furthermore, note that choosing investment is the same thing as choosing tomorrowís capital. In other words, any target level of capital tomorrow is consistent with one and only one level of investment today, and once investment today is determined the capital stock tomorrow is implicitly determined as well.

" The householdís budget constraints can be combined to yield a single constraint:

ct + kt+1 & (1& .)kt| {z } =It

' wtnt +Rtkt +!t.

Given this single constraint, the householdís problem involves choosing todayís con- sumption, todayís labor supply, and tomorrowís capital stock (todayís capital stock is inherited from the past, so nothing can be done about this variable within periods) in order to maximize lifetime utility. In particular, we assume that the house- holdís problem is:

max ct;nt;kt+1

Ut = Et X1

t=0 *t [u(ct)+v (1&nt)]

where: Et is the (conditional on the information available at time t) expectation op- erator, which is necessary given that in the modeling framework there is uncertainty (details follow below); Ut is lifetime utility; u(ct) + v (1&nt) is instantaneous utility (u and v are functions that satisfy: u0 > 0; u00 < 0; v0 > 0; and v00 < 0); such that:

ct +kt+1 & (1& .)kt ' wtnt +Rtk +!t.

Note the use of the summation operator in the objective function, which is there because of the assumption of inÖnitely lived agents. In addition, note that in this context the household is essentially able to ìsaveî via capital accumulation. We could havealsoassumedthat the household is able to save viamore typical (fromthe intuitive standpoint) means, such as bond holdings, but when the assumption is made that the household owns the capital stock no other savings means are generally assumed as additional savings means yield the same bottom-line results. (We will go over the case of bond holdings when we explore the setup in which the Örm owns the capital stock.)

" The current value Lagrangian is:

L = Et X1

t=0 *tfu(ct)+v (1&nt)

+4t [wtnt +Rtkt +!t & ct &kt+1 +(1& .)kt]g.

This formulation is called ìcurrent valueî because, as will become clear below, the Lagrange multiplier 4t (4 is the Greek letter ìlambdaî) is the marginal utility of con- sumption at time t. An alternative formulation is called the present value Lagrangian, which is stated as follows:

L = Et X1

t=0 f*t [u(ct)+v (1&nt)]

+4t [wtnt +Rtkt +!t & ct &kt+1 +(1& .)kt]g,

i.e., only instantaneous utility is discounted. Both approaches yield the same bottom- line results, and the only di§erence lies in the interpretation of the Lagrange multiplier. (Importantly, note that both approaches require an inÖnite sum given the assumption that agents are inÖnitely lived.) Throughout the class, we will stick to the cur- rent value formulation as it is the most widely used (so, unless otherwise noted, whenever I refer to a Lagrangian I mean itís current value formula- tion). Returning to this formulation, the Örst-order conditions are:

@L @ct

! = 0 for each t

! Et* tu0 (ct)&Et*

t4t = 0

! Etu 0 (ct)&Et4t = 0

(dividing through by *t; we can do this

since * is a known parameter)

u0 (ct) = 4t

(time t variables are known, so the expectation

operator is not needed for time-t indexed variables

once the Örst order conditions are taken);

@L @nt

! = 0 for each t

! &*tEtv 0 (1&nt)+*

tEt4twt = 0 (using the calculus chain rule)

! &Etv 0 (1&nt)+Et4twt = 0

(dividing through by *t; we can do this

since * is a known parameter)

! v0 (1&nt) = 4twt (time t variables are known, so the expectation

operator is not needed for time-t indexed variables

once the Örst order conditions are taken);

@L @kt+1

! = 0 for each t+1

! &Et* t4t +Et*

t+14t+1 [Rt+1 +(1& .)] = 0 ! &Et4t +Et*

t4t+1 [Rt+1 +(1& .)] = 0 (dividing through by *t; we can do this

since * is a known parameter)

! 4t = *Et4t+1 [Rt+1 +(1& .)] (time t variables are known, so the expectation

operator is not needed for time-t indexed variables once

the Örst order conditions are taken, but it is needed

for any time periods further ahead since, by assumption,

there is uncertainty about the future);

and

@L @4t

! = 0 for each t

! Et* t [wtnt +Rtkt +!t & ct &kt+1 +(1& .)kt] = 0

! Et* t [wtnt +Rtkt +!t & ct &kt+1 +(1& .)kt] = 0 (dividing through by *t; we can do this

since * is a known parameter)

! ct +kt+1 & (1& .)kt = wtnt +Rtk +!t (time t variables are known, so the expectation

operator is not needed for time-t indexed variables once

the Örst order conditions are taken).

" Some notes:

ñ The Örst two Örst-order conditions should, at this stage in the class, be unprob- lematic. Letís take a closer look at the third one, though, which is a situation in which intertemporal optimization does become more intricate compared to static optimization. In essence, when the household looks at its Lagrangian it is really looking at it by expanding the inÖnite sum. In other words,

L = Et X1

t=0 f*t [u(ct)+v (1&nt)]

+4t [wtnt +Rtkt +!t & ct &kt+1 +(1& .)kt]g

is generically equivalent to (by ìgenericallyî I mean assuming that the optimiza-

tion problem begins in any arbitrary period t)

L = Etf* t [u(ct)+v (1&nt)]

+4t [wtnt +Rtkt +!t & ct &kt+1 +(1& .)kt] +*t+1 [u(ct+1)+v (1&nt+1)]

+4t+1 [wt+1nt+1 +Rt+1kt+1 +!t+1 & ct+1 &kt+2 +(1& .)kt+1] +*t+2 [u(ct+2)+v (1&nt+2)]

+4t+2 [wt+2nt+2 +Rt+2kt+2 +!t+2 & ct+2 &kt+3 +(1& .)kt+2] +:::

So, upon taking the Örst order condition for ct, we have to account for everywhere that ct appears in this inÖnite Lagrangian: the boxed cts below:

L = Etf* t h u ' ct

( +v (1&nt)

+4t

h wtnt +Rtkt +!t & ct &kt+1 +(1& .)kt

+*t+1 [u(ct+1)+v (1&nt+1)] +4t+1 [wt+1nt+1 +Rt+1kt+1 +!t+1 & ct+1 &kt+2 +(1& .)kt+1]

+*t+2 [u(ct+2)+v (1&nt+2)] +4t+2 [wt+2nt+2 +Rt+2kt+2 +!t+2 & ct+2 &kt+3 +(1& .)kt+2]

+:::

Similarly, upon taking the Örst order condition for nt, we have to account for everywhere that nt appears in this inÖnite Lagrangian: the boxed nts below:

L = Etf* t h u(ct)+v

' 1& nt

+4t

h wt nt +Rtkt +!t & ct &kt+1 +(1& .)kt

+*t+1 [u(ct+1)+v (1&nt+1)] +4t+1 [wt+1nt+1 +Rt+1kt+1 +!t+1 & ct+1 &kt+2 +(1& .)kt+1]

+*t+2 [u(ct+2)+v (1&nt+2)] +4t+2 [wt+2nt+2 +Rt+2kt+2 +!t+2 & ct+2 &kt+3 +(1& .)kt+2]

+:::

Akin to the preceding, upon taking the Örst order condition for kt+1, we have to account for everywhere that kt+1 appears in this inÖnite Lagrangian: the boxed

kt+1s below:

L = Etf* t [u(ct)+v (1&nt)]

+4t

h wtnt +Rtkt +!t & ct & kt+1 +(1& .)kt

+*t+1 [u(ct+1)+v (1&nt+1)]

+4t+1

h wt+1nt+1 +Rt+1 kt+1 +!t+1 & ct+1 &kt+2 +(1& .) kt+1

+*t+2 [u(ct+2)+v (1&nt+2)] +4t+2 [wt+2nt+2 +Rt+2kt+2 +!t+2 & ct+2 &kt+3 +(1& .)kt+2]

+:::

Finally, upon taking the Örst order condition for 4t, we have to account for every- where that 4t appears in this inÖnite Lagrangian: the boxed 4t below:

L = Etf* t [u(ct)+v (1&nt)]

+ 4t [wtnt +Rtkt +!t & ct &kt+1 +(1& .)kt]

+*t+1 [u(ct+1)+v (1&nt+1)] +4t+1 [wt+1nt+1 +Rt+1kt+1 +!t+1 & ct+1 &kt+2 +(1& .)kt+1]

+*t+2 [u(ct+2)+v (1&nt+2)] +4t+2 [wt+2nt+2 +Rt+2kt+2 +!t+2 & ct+2 &kt+3 +(1& .)kt+2]

+:::

ñ Importantly, notice thatbyeachÖrstordercondition I clariÖedthat itwas relevant ìfor each t.î You may be wondering why. Well, the issue is that conceptually this intertemporal problem is thought of as choosing a stream of consumption, labor, and capital conditional on the information available at time t. In other words, at each time t you observe the state of the economy and choose the optimal values of consumption, labor, and capital for the entire future conditional on that information. Of course, next period when new information arrives (or not) you can reoptimize and choose a new ìpathî for each of these variables.

ñ Finally, note that consumption and labor can be chosen any period no matter what. But, because there isaone-period lagtobuildas farascapital accumulation goes, todayís capital is a variable that is inherited from the past and cannot be modiÖed instantaneously. Only future capital can be modiÖed. As such, capital is referred to as a state (endogenous) variable.

" With all that said and done, letís proceed to clean up the optimality conditions. Com- bine the Örst and second Örst-order conditions to obtain:

v0 (1&nt) = u 0 (ct)wt. (1)

In turn, combine the Örst and third Örst-order conditions to obtain:

u0 (ct)| {z } =+t

= *Etu 0 (ct+1)| {z } =+t+1

(Rt+1 +(1& .)) , (2)

which is calledtheìEuler equation.î Inaddition, weneedwhatís calledtheìtransver- sality condition:î

lim t!1

*tkt+1u 0 (ct) = 0; (3)

Combined with the Örst-order conditions from earlier the transversality condition is necessary and su¢cient for an optimum.

ñ The transversality condition is most easily understood by supposing that the problem is Önite. At the end of time, you would never want to leave anything on the table, so to speak. If time ended in period T, then the transversality condition says that

*TkT+1u 0 (cT) = 0.

Note that * is always positive and u0 is always positive as well (ìmore is betterî of an economic good). So, the only way that this transversality condition can hold is if kT+1 = 0. Why? There is no sense in kT+1 > 0. It could be consumed in the prior period T, therefore enhancing the end-of time utility. (And yes, the capital stock can be consumedójust think of disinvestment from which the proceeds are credited towards the consumption good). All told, if the present value of kT+1u0 (cT) is not zero, then the household has ìover-savedî and could not beoptimizing. There is a transversality condition for each endogenous state variable in any one problem.

" Before moving along, letís delve deeper into the interpretation of the Euler equation. To do so, letís suppose momentarily that instead of there being inÖnite periods there were only 2 and also that there is no uncertainty, meaning that we can omit the use of expectation operator. Then, instantaneous utility is given by:

Ut = X1

t=0 *t [u(ct)+v (1&nt)]

= *0 [u(c0)+v (1&n0)]+* 1 [u(c1)+v (1&n1)]

= u(c0)+v (1&n0)+* [u(c1)+v (1&n1)] :

ñ Letís take the total derivative:

dUt = u 0 0 ,dc0 &v

0 0 ,dn0 +*u

0 1 ,dc1 &*v

0 1 ,dn1.

and recall that u0t is the marginal utility of consumption in period t. So, for dct = #, where # is some number, u0t , # gives the extra utility of consuming # in period t. Similarly, &v0t is the marginal disutility of work in period t, so for dnt = #, where # is some number, &v0t ,# gives the extra disutility of working # in period t. In particular, then, *u01 ,# tells us how much the household values in

period 0 the extra utility obtained fromconsuming # in period 1 (this extra utility is discounted by the subjective factor *). Now, letís assume that dn0 = dn1 = 0 in which case the preceding equation becomes:

dUt = u 0 0dc0 +*u

0 1dc1

! dUt u00

= dc0 + *u01 u00 dc1.

The left hand side of this expression is now giving us the change in lifetime utility givena reallocationof consumptionacross periods relative towhat this extravalue of consumption is worth to the household at the beginning of the optimization problem, or, alternatively, the change in lifetime utility in terms of the extra value of utility at the beginning of the optimization problem. Suppose that the household decides to transfer 1 unit of consumption from period 0 to period 1 by saving via investment. Then, dc0 = &1 and dc1 = 1 + R1 & . (the household obtains e§ective interest R1 &. from saving 1 unit of consumption in period 0 so in period 1 it gets back 1 + R1 & . units of consumption). Then, the preceding equation becomes:

dUt u00

= &1+ *u01 u00 (1+R1 & .).

When the household has allocated consumption optimally across periods, then by optimality there is no reallocation of consumption that can provide additional utility. Therefore, given an optimal allocation of consumption across periods, it must be the case that dUt = 0. Thus, the preceding equation implies that at an optimum:

0 = &1+ *u01 u00 (1+R1 & .)

! u00 = *u 0 1(1+R1 & .),

and, more generally, between any two periods t and t+1:

u0t = *u 0 t+1(1+Rt+1 & .)

= *u0t+1 (Rt+1 +(1& .)) ,

which is the Euler equation. So, that the Euler equation is satisÖed means that indeed the household is at an optimum.

2.1.2 The Firm

" The Örm produces (Önal) output yt per the production function

yt = ztf (kt;nt) ,

where: zt is an exogenous productivity process and f is a function that takes as inputs capital and labor. As such, the Örm hires labor and capital to produce.

(The Örm can choose to use any amount of the economyís capital stock that is available today). We assume fk > 0, fn > 0, fkk < 0 and fnn < 0. Revenue each period is simply equal to output (recall that the price of Önal goods is normalized to 1). All told, the Örmís problem is:

max kt;nt

!t = ztf (kt;nt)&wtnt &Rtkt| {z } =(t

(no expectation operator needed

because all time t variables are known),

where the Örm takes as given the wage rate and the rental rate as well as the exogenous productivity processóuncertainty about the evolution of productivity is where the whole frameworkís uncertainty stems from (further details below).

" The Örst-order conditions are:

@!t @nt

! = 0 ! ztfn (kt;nt)&wt = 0

! ztfn (kt;nt) = wt (4) (MPLt = wt)

and

@!t @kt

! = 0 ! ztfk (kt;nt)&Rt = 0

! ztfk (kt;nt) = Rt (5) (MPKt = Rt)

NOTE: because the Örm is choosing how much of todayís available capital stock to use, there is no Örst-order condition for kt+1 as far as the it is concerned.

2.1.3 Closing the Model

" To close the model we need to make assumptions on the exogenous productivity process zt, which is the only exogenous variable in the model. In particular, we specify a stochastic productivity process. We assume that this process is well-characterized following a mean zero AR(1) in the natural log:

lnzt = ? lnzt$1 +" z t ,

where the exogenously determined constant parameter ? 2 (0;1) (? is the Greek letter ìrhoî) and "zt is an ìexogenous productivity shockî (" is the Greek letter ìepsilonî). In steady state (more details later) "zt = 0 8t, and all variables have the same value across time. Therefore, in steady state the productivity process from above implies that

lnz = ? lnz ! z = 1.

ñ Note that nothing can be done about the exogenous productivity process as far as the optimizing agents in the economy are concerned. So, you can think of z as being an exogenous state variable. As such, in the present context the ìstate of the economyî is summarized by the level of productivity and the level of capital in any given period. In terms of timing, please note that the implicit assumption is that any one periodís level of productivity is revealed before economic agents make decisions. So, all time t decisions are made given knowledge of zt but at the same time conditional on uncertainty regarding future levels of zt.

2.1.4 The Bottom Line: Key Equations

" We now have, combining (1) and (4):

v0 (1&nt) = u 0 (ct)ztfn (kt;nt)| {z }

=wt

(obtained by substituting out the wage

in the household FOCs using MPLt);

combining (2) with (5):

u0 (ct) = *Etu 0 (ct+1)

6 4zt+1fk (kt+1;nt+1)| {z }

=Rt+1

+(1& .)

7 5

(by substituting out Rt+1 through the forward iteration of MPKt,

that is, ztfk (kt;nt) = Rt ! zt+1fk (kt+1;nt+1) = Rt+1);

kt+1 = It +(1& .)kt;

and also: ct + It = wtnt +Rtkt +!t.

Note that these 4 equations can be thought of as implicitly solving for the 4 variables: c, I, n, and k.

2.1.5 Equilibrium

" In a competitive general equilibrium: (1.) households optimize; (2.) Örms optimize; (3.) markets clear. In the present case, a competitive equilibrium is characterized by:

1. A set of prices (wt,Rt) and allocations (ct,nt,kt+1,It) taking kt as well as the stochastic process for zt as given;

2. All optimality conditions hold (that is, conditions (1), (2), (4), and (5));

3. The labor, capital, goods markets clear (ndt = n s t, k

d t = k

s t, where d

denotes demand and s denotes supply, and yt = ct + It, respectively);

4. The transversality condition holds.

2.2 Centralized/Planning Setup

" AÖnalwayof solvingall of thepreceding isassumingthataìbenevolent socialplannerî is in charge of the entire economyís optimization problem. In this case, the entire economyís problem can be cast as

max ct;nt;kt+1

Ut = Et X1

t=0 *t [u(ct)+v (1&nt)]

such that: ct + It = yt;

yt = ztf (nt;kt) ;

and kt+1 = It +(1& .)kt.

As before, we also assume that

lnzt = ? lnzt$1 +" z t .

Note that the planner does not care about prices, such as wages and rental rates, because technically they donít really exist in this centrally planned economy. The planner is distributing everything around optimally and without need of prices because the planner is in control of everything. ìStu§ is just produced by Örms and given to households directly.î In e§ect, there is technically no ìmarketî per say when the economy is being centrally planned. As before, we can restate the preceding problem by combining constraints and stating:

max ct;nt;kt+1

Ut = Et X1

t=0 *t [u(ct)+v (1&nt)]

such that: ct +kt+1 & (1& .)kt = ztf (nt;kt) .

(We assume the same productivity process as before).

" The current value Lagrangian is:

L = Et X1

t=0 *tfu(ct)+v (1&nt)+4t [ztf (nt;kt)& ct &kt+1 +(1& .)kt]g.

" The Örst-order conditions are:

@L @ct

! = 0 ! Etu

0 (ct)&Et4t = 0

! u0 (ct) = 4t (time t variables are known);

@L @nt

! = 0

! &Etv 0 (1&nt)+Et4tztfn (nt;kt) = 0

(using the calculus chain rule)

! v0 (1&nt) = 4tztfn (nt;kt) ;

@L @kt+1

! = 0

! &Et* t4t +Et*

t+14t+1 [zt+1fk (nt+1;kt+1)+(1& .)] = 0 ! 4t = *Et4t+1 [zt+1fk (nt+1;kt+1)+(1& .)] ;

and

@L @4t

! = 0

! ct +kt+1 & (1& .)kt = ztf (nt;kt) .

And, the transversality condition is:

lim t!1

*tkt+1u 0 (ct) = 0.

" So, now we have 4 equations in 4 unknowns (c, I, n, and k):

v0 (1&nt) = u 0 (ct)ztfn (nt;kt)

(follows by substituting out 4t in

the Örst two Örst-order conditions);

u0 (ct) = *Etu 0 (ct+1) [zt+1fk (nt+1;kt+1)+(1& .)]

(obtained by combining the Örst

and third Örst-order conditions);

ct +kt+1 & (1& .)kt = ztf (nt;kt) ;

and ct + It = ztf (nt;kt) .

" These conditions are equivalent to the bottom line equations the decentral- ized case presented earlier.

3 Analytical Steady States

" Consider an economy characterized by the household problem:

max ct;nt;kt+1

Et X1

t=0 *t [ln(ct)+B ln(1&nt)]

such that ct + It ' wtnt +Rtkt +!t;

and kt+1 = It +(1& .)kt,

where B (the Greek letter ìthetaî) is a scalar, and the Örm problem:

max nt;kt

!t = ztk 0 t n

1$0 t &wtnt &Rtkt,

where C (the Greek letter ìalphaî) is a parameter such that 0 < C < 1. Moreover, the exogenous productivity process follows an AR(1) process in natural logs:

lnzt = ? lnzt$1 +" z t .

As in the development thus far, the price of the Önal good is normalized to 1 and all non-price variables are normalized by the population, which consists of a unit mass.

" The Lagrangian associated with the household problem is:

L = Et X1

t=0 *tf(ln(ct)+B ln(1&nt))

+4t (wtnt +Rtkt +!t & ct &kt+1 +(1& .)kt)g,

and the Örst-order conditions are:

@L @ct

! = 0

! 1

ct = 4t;

@L @nt

! = 0

! B

1&nt = 4twt;

@L @kt+1

! = 0

! 4t = *Et4t+1 [Rt+1 +(1& .)] ;

and

@L @4t

! = 0

ct + It = wtnt +Rtkt +!t.

In addition, the relevant transversality condition is:

lim t!1

*tkt+1 1

ct = 0.

" In turn, the Örmís Örst-order conditions are:

@!t @nt

! = 0

! (1&C)ztk 0 t n

$0 t = wt;

and

@!t @kt

! = 0

! Cztk 0$1 t n

1$0 t = Rt.

" Except under very special cases there are no analytical solutions for models involving intertemporal optimization such as the RBC models weíve studied. However, we can analytically characterize the solution for a special case in which the variables of the model are constant. This is called the steady state. In steady state, for any variable x, xt = xt+j for j 2 Z (and given this situation, of course, no expectation operators are applicable). We will now proceed to derive a set of equations that fully characterize the steady state of this economy in closed form solution.

ñ Combining the householdís Örst-order condition for the capital stock with the Örmís Örst-order condition for capital implies that

4t = *Et4t+1 0 Czt+1k

0$1 t+1 n

1$0 t+1 +(1& .)

In steady state

4 = *4 0 Czk0$1n1$0 +(1& .)

! 1 = * 2 Ck0$1n1$0 +(1& .)

3 ,

where the last line follows fromtheAR(1)process forproductivity lnzt = ? lnzt$1+

"zt , which in steady state (" z t = 0 8t) implies that:

lnz = ? lnz

! lnz (1&?) = 0 ! lnz = 0 ! z = 1.

ñ Of note, within this context Önding k=n is the easiest way to Önd steady-state values for all other endogenous variables. So, moving along from the preceding, we have:

1 = * 2 Ck0$1n1$0 +(1& .)

! 1 = *

5 k

60$1 +(1& .)

! 1

* & (1& .) = C

5 k

60$1

! 5 k

60$1 =

1 2 & (1& .) C

! k

n =

1 2 & (1& .) C

!1=(0$1)

! k

n =

1 2 & (1& .)

!1=(1$0) .

ñ Now, letís turn to Önding equations for the wage and the rental rate. From the Örmís Örst order condition for labor, in steady state:

w = (1&C)zk0n$0t

! w = (1&C)z 5 k

! w = (1&C)

1 2 & (1& .)

!0=(1$0) ,

where the last line follows from substituting the steady state capital-labor ratio from before and the fact that in steady state z = 1. Using analogous reasoning, it is straightforward to show that:

R = C

1 2 & (1& .)

!$1 .

ñ Now, letís turn to investment and labor. From the capital accumulation equation

kt+1 = It +(1& .)kt,

in steady state

k = I +(1& .)k ! I = .k.

In equilibrium combining the householdís constraint with the Örmís objective function we obtain the accounting identity

yt = ct + It,

which in steady state and given our assumptions on the production function im- plies that

k0n1$0 = c+ I

! 5 k

60 n = c+ .k,

where weíve used the investment steady-state level. Dividing by n and rearrang- ing, the preceding implies that

n =

5 k

60 & .

5 k

6 . (*)

Recall that the householdís Örst-order conditions for consumption and leisure imply that

ct = 4t

and B

1&nt = 4twt.

Combining yields B

1&nt = wt ct .

Eliminate the wage by substituting in the Örmís Örst-order condition for labor:

1&nt = (1&C)ztk0t n

$0 t

! B

1&nt = (1&C)zt

' kt nt

! ct = (1&nt) (1&C) B

5 kt nt

! ct nt =

5 1&nt nt

6 1&C B

5 kt nt

! c

n =

5 1&n n

6 1&C B

5 k

60 , (**)

where the second-to last equation follows from dividing through by nt and the last expression must hold in steady state. Now, use equations (*) and (**) to eliminate the consumption-labor ratio:

5 1&n n

6 1&C B

5 k

60 =

5 k

60 & .

5 k

! (1&n) 1&C B

5 k

60 = n

55 k

60 & .

5 k

! 1&C B

5 k

60 = n

55 k

60 + 1&C B

5 k

60 & .

5 k

! 1&C B

5 k

60 = n

5 B +1&C

5 k

60 & .

5 k

! n = 1$0 4

2 k n

4+1$0 4

2 k n

30 & .

2 k n

and therefore

n =

1$0 4

5 0

1 " $(1$5)

60=(1$0)

4+1$0 4

5 0

1 " $(1$5)

60=(1$0) & .

5 0

1 " $(1$5)

61=(1$0)

follows from substituting in the expression for the steady state capital-labor ratio derived earlier.

ñ Focusing on output, we have

yt = ztk 0 t n

1$0 t ,

and therefore, in steady state y = k0n1$0.

ñ Moving along to consumption, in equilibrium combining the householdís con- straint with the Örmís objective function we obtain the accounting identity

yt = ct + It,

which in steady state and given our assumptions on the production function im- plies that

k0n1$0 = c+ I

! 5 k

60 n = c+ .k

! c = 5 k

60 n& .

5 k

6 n

! c = n 85 k

60 & .

5 k

69 .

" All told, the steady state equations for z, n, k, I, c, w, and R, are, respectively,

z = 1;

n =

1$0 4

5 0

1 " $(1$5)

60=(1$0)

4+1$0 4

5 0

1 " $(1$5)

60=(1$0) & .

5 0

1 " $(1$5)

61=(1$0) ;

n =

1 2 & (1& .)

!1=(1$0)

! k = n

1 2 & (1& .)

!1=(1$0)

! k =

1$0 4

5 0

1 " $(1$5)

60=(1$0)

4+1$0 4

5 0

1 " $(1$5)

60=(1$0) & .

5 0

1 " $(1$5)

61=(1$0)

1 2 & (1& .)

!1=(1$0)

! k =

1$0 4

5 0

1 " $(1$5)

60=(1$0)

4+1$0 4

5 0

1 " $(1$5)

60=(1$0) & .

5 0

1 " $(1$5)

61=(1$0)

1 2 & (1& .)

!1=(1$0) ;

I = .k

! I = . 1$0 4

4+1$0 4

5 0

1 " $(1$5)

60=(1$0) & .

5 0

1 " $(1$5)

61=(1$0) ;

c = n

55 k

60 & .

5 k

! c =

1$0 4

5 0

1 " $(1$5)

60=(1$0)

4+1$0 4

5 0

1 " $(1$5)

60=(1$0) & .

5 0

1 " $(1$5)

61=(1$0)

8 >>><

>>>:

"5 0

1 " $(1$5)

61=(1$0)#0

&. 5

0 1 " $(1$5)

61=(1$0)

9 >>>=

>>>; ;

w = (1&C)

1 2 & (1& .)

!0=(1$0) ;

and

R = C

1 2 & (1& .)

!$1 .

" We will make extensive use of steady states when programming DSGE models using Dynare and Matlab.